Geoscience Reference
In-Depth Information
To compute the pooled within-group variance, we first get the corrected sum of squares (
SS
)
within each group:
−
(
)
=+++ −
2
∑
X
n
2
()
99
11
A
∑
2
2
2
2
…
SS
=
X
11
8
11
=
34
A
A
A
−
(
)
=+++−
2
∑
X
n
2
()
88
11
B
∑
2
2
2
2
SS
=
X
99
…
6
=
54
B
B
B
Then the pooled variance is
SS
+
−+ −
==
SS
88
20
2
A
B
s
=
44
.
(
n
1
)
(
n
1
)
A
B
Hence,
90
.
−
80
.
10
0 800000
.
t
=
=
= 1 118
.
.
11 11
11 11
+
44
.
( ()
This value of
t
has (
n
A
- 1) + (
n
B
- 1) degrees of freedom. If it exceeds the tabular value (from a
distribution of
t
table) at a specified probability level, we would reject the hypothesis. The dif-
ference between the two means would be considered significant (larger than would be expected
by chance if there is actually no difference). In this case, tabular
t
with 20 degrees of freedom at
the 0.05 level is 2.086. Since our sample value is less than this, the difference is not significant
at the 0.05 levels.
One of the unfortunate aspects of the
t
test and other statistical methods is that almost any kind of
numbers can be plugged into the equations. But, if the numbers and methods of obtaining them do
not meet certain requirements, then the result may be a fancy statistical facade with nothing behind
it. In a handbook of this scope it is not possible to make the reader aware of all of the niceties of
statistical usage, but a few words of warning are certainly appropriate.
A fundamental requirement in the use of most statistical methods is that the experimental mate-
rial be a random sample of the population to which the conclusions are to be applied. In the
t
test
of white pine races, the plots should be a sample of the sites on which the pine are to be grown, and
the planted seedlings should be a random sample representing the particular race. A test conducted
in one corner of an experimental forest may yield conclusions that are valid only for that particular
area or sites that are about the same. Similarly, if the seedlings of a particle race are the progeny of
a small number of parents, their performance may be representative of those parents only, rather
than of the race.
In addition to assuming that the observations for a given race are a valid sample of the population
of possible observations, the
t
test described above assumes that the population of such observations
follows the normal distribution. With only a few observations, it is usually impossible to determine
whether or not this assumption has been met. Special studies can be made to check on the distribu-
tion, but often the question is left to the judgment and knowledge of the research worker.
Finally, the
t
test of unpaired plots assumes that each group (or treatment) has the same popula-
tion variance. Since it is possible to compute a sample variance for each group, this assumption
can be checked with Bartlett's test for homogeneity of variance. Most statistical textbooks present
variations of the
t
test that may be used if the group variances are unequal.
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